Isotopic equivalence by Bézier curve subdivision for application to high performance computing

• We establish ambiently isotopic control polygons for Bézier curves via subdivision.
• We provide closed-form formulas for the required number of subdivision iterations.
• The results are being applied in computer animation.

For an arbitrary degree Bézier curve BB, we first establish sufficient conditions for its control polygon to become homeomorphic to BB via subdivision. This is extended to show a subdivided control polygon that is ambient isotopic to BB. We provide closed-form formulas to compute the corresponding number of iterations for equivalence under homeomorphism and ambient isotopy. The development of these a priori values was motivated by application to high performance computing (HPC), where providing estimates of total run time is important for scheduling.